We’ve mentioned before that you can estimate π by dropping needles on the floor. (Reader Steven Karp also directed me to this remarkable solution, from Daniel A. Klain and Gian-Carlo Rota’s *Introduction to Geometric Probability* [1997].)

Here’s a related curiosity. If a circle of diameter *L* is placed at random on a pattern of circles of unit diameter, which are arranged hexagonally with centers *C* apart, then the probability that the placed circle will fall entirely inside one of the fixed circles is

If we put *k* = *C*/(1 – *L*), we get

And a frequency estimate of *P* will give us an estimate of π.

Remarkably, in 1933 A.L. Clarke actually tried this. In *Scripta Mathematica*, N.T. Gridgeman writes:

His circle was a ball-bearing, and his scissel a steel plate. Contacts between the falling ball and the plate were electrically transformed into earphone clicks, which virtually eliminated doubtful hits. With student help, a thousand man-hours went into the accumulation of *N* = 250,000. The *k* was about 8/5, and the final ‘estimate’ of π was 3.143, to which was appended a physical error of ±0.005.

“This is more or less the zenith of accuracy and precision,” Gridgeman writes. “It could not be bettered by any reasonable increase in *N* — even if the physical error could be reduced, hundreds of millions of falls would be needed to establish a third decimal place with confidence.”

(N.T. Gridgeman, “Geometric Probability and the Number π,” *Scripta Mathematica* 25:3 [November 1960], 183-195.)